几个著名积分不等式的等价性

184Vol.18,No.4200512BASICSCIENCESJOURNALOFTEXTILEUNIVERSITIESDec.,200533:100628341(2005)0420404203(,726000)X:Diaz2MetcalfCauchy2SchwarzKantorovich,,.:Diaz2Metcalf;Cauchy2Schwarz;Kantorovich;;:O178:A[13]Diaz2MetcalfCauchy2SchwarzKantorovich,[4]..1,.1(Diaz2Metcalf)f(x)g(x)[a,b],[a,b]f(x)0,mg(x)öf(x)M,ba(g(x))2dx+mMba(f(x))2dx(m+M)baf(x)g(x)dx,(1)g(x)=mf(x)g(x)=Mf(x).2(Cauchy2Schwarz)f(x)g(x)[a,b],[a,b]0m1f(x)M1,0m2g(x)M2,m2M2m1M1ba(f(x))2dx+m1M1m2M2ba(g(x))2dxM1M2m1m2+m1m2M1M2baf(x)g(x)dx,(2)f(x)g(x)=M1m2f(x)g(x)=m1M2.3(Kantorovich)f(x),g(x)1ög(x)[a,b],[a,b]0mg(x)M,f(x)0,baf(x)g(x)dxbaf(x)g(x)dx(m+M)24mMbaf(x)dx2,(3)X:2005206213:(EHA030431):(19602),,,,.E2mail:[e-mail](x)=mg(x)=M.23:1,1]2]3,3]1.f(x)g(x)[a,b],,[a,b]n$i=[xi-1,xi],i=1,2,,n,Ni[xi-1,xi],$xi=b-an.mg(Ni)f(Ni)M,f(Ni)0,g(Ni)f(Ni)-mM-g(Ni)f(Ni)0,g(Ni)f(Ni)-mM-g(Ni)f(Ni)(f(Ni))20,(m+M)f(Ni)g(Ni)-(g(Ni))2-mM(f(Ni))20,(m+M)f(Ni)g(Ni)(g(Ni))2+mM(f(Ni))2.(m+M)6ni=1f(Ni)g(Ni)õ$xi6ni=1(g(Ni))2õ$xi+mM6ni=1(f(Ni))2õ$xi.$xi0(n),(m+M)baf(x)g(x)dxba(g(x))2dx+mMba(f(x))2dx.,g(x)=mf(x)g(x)=Mf(x).(1)(1),m=m2öM1,M=M2öm1,M2m1+m2M1baf(x)g(x)dxba(g(x))2dx+m2M1õM2m1ba(f(x))2dx.(4)(4),(2),1]2.(2)(2),M2öm1=M,m2öM1=m,(f(x))2=h(x)k(x),(g(x))2=h(x)ök(x),h(x)=f(x)g(x),k(x)=f(x)ög(x),h(x)0,0mk(x)M.bah(x)k(x)dx+mMbah(x)k(x)dx(m+M)bah(x)dx.bah(x)k(x)dx+mMbah(x)k(x)dx2mMbah(x)k(x)dxbah(x)k(x)dx1ö2.2mMbah(x)k(x)dxbah(x)k(x)dx1ö2(m+M)bah(x)dx.(5)(5),bah(x)k(x)dxõbah(x)k(x)dx(m+M)24mMbah(x)dx2.(6)(6)Kantorovich,2]3.(1),(2)2]3.f(x),g(x)1ög(x)[a,b].,[a,b]n$i=[xi-1,xi],i=1,2,,n,Ni[xi-1,xi],$xi=(b-a)ön.(g(Ni)-m)(g(Ni)-M)0,f(Ni)0,(g(Ni)-m)(g(Ni)-M)f(Ni)0,f(Ni)g(Ni)+mMf(Ni)g(Ni)(m+M)f(Ni).6ni=1(f(Ni)g(Ni))õ$xi+mM6ni=1f(Ni)g(Ni)õ$xi(m+M)6ni=1f(Ni)õ$xi.$xi0(n),,baf(x)g(x)dx+mMbaf(x)g(x)dx(m+M)baf(x)dx.(7)(7)Kantorovich.(7),,2baf(x)g(x)dxmMbaf(x)g(x)dx1ö2(m+M)baf(x)dx.(8)(8),(3).(7),f(x)=h(x)k(x),g(x)=h(x)ök(x),5044ba(h(x))2dx+mMba(k(x))2dx(m+M)bah(x)k(x)dx.(9)(9)Diaz2Metcalf,13,23.,1Z2Z3.:[1]POLYAG,SZEGObG.()[M].,,.:,1981.[2].[M].:,2003.[3].[M].3.:,2004.[4].Kantorovich[J].(),2004,24(4):2652267.EquivalenceofseveralfamouslintegralinequalityQIAOXi2min(Dept.ofMath.,ShangluoTeachersCollege,Shanluo,Shaanxi726000,China)Abstract:OnthebaseofstudyingfamousDiaz2Metcalfpopularizedintegralinequality,Cauchy2SchwarzreinforcedintegralinequalityandKantorovichreinforcedintegralinequality,itistodemonstratetheequivalencebetweenthemwithconstructivemethodthroughinductionandanalogy.Keywords:Diaz2Metcalfintegralinequality;Cauchy2Schwarzintegralinequality;Kantorovichintegralinequality;equivalence;constructivemethod:;:MR(MR)(AMS),2005,MR170,7,5(1)MR2004,48,1MR(ISSN)ö2003200410068341200348()1000274X198236025()100138571995190220253987X198236317()100712611995193060418